Abstract
We prove the existence of the critical fixed point (F, G) for MacKay’s renormalization operator for pairs of maps of the plane. The maps F and G commute, are area-preserving, reversible, real analytic, and they satisfy a twist condition.
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Kadanoff L.P.: Scaling for a critical Kolmogorov–Arnold–Moser trajectory. Phys. Rev. Lett. 47, 1641–1643 (1981)
MacKay, R.S.: Renormalisation in Area Preserving Maps. Thesis, Princeton, 1982, London: World Scientific, 1993
MacKay R.S.: Renormalisation approach to invariant circles in area–preserving maps. Physica D 7, 283–300 (1983)
Greene J.M., Mao J.-M.: Higher-order fixed points of the renormalisation operator for invariant circles. Nonlinearity 3, 69–78 (1990)
Wilbrink J.: New fixed point of the renormalisation operator associated with the recurrence of invariant circles in generic Hamiltonian maps. Nonlinearity 3, 567–584 (1990)
Stirnemann A.: Renormalization for golden circles. Commun. Math. Phys. 152, 369–431 (1993)
Stirnemann A.: Towards an existence proof of MacKay’s fixed point. Commun. Math. Phys. 188, 723–735 (1997)
Escande D.F., Doveil F.: Renormalisation method for computing the threshold of the large scale stochastic instability in two degree of freedom Hamiltonian systems. J. Stat. Phys. 26, 257–284 (1981)
Mehr A., Escande D.F.: Destruction of KAM Tori in Hamiltonian systems: link with the destabilization of nearby cycles and calculation of residues. Physica D 13, 302–338 (1984)
Chandre C., Jauslin H.R.: Renormalization–group analysis for the transition to chaos in Hamiltonian systems. Phys. Rep. 365, 1–64 (2002)
Koch, H.: Renormalization of vector fields. In: Holomorphic Dynamics and Renormalization, Lyubich, M., Yampolsky, M. (eds.), Fields Institute Communications, Providence, RI: Amer. Math. Soc. 2008, pp. 269–330
Koch H.: On the renormalization of Hamiltonian flows, and critical invariant tori. Disc. Cont. Dyn. Sys. A 8, 633–646 (2002)
Koch H.: A Renormalization group fixed point associated with the breakup of golden invariant Tori. Disc. Cont. Dyn. Sys. 11, 881–909 (2004)
Koch H.: Existence of critical invariant tori. Erg. Theor. Dyn. Syst. 28, 1879–1894 (2008)
Arioli, G., Koch, H.: The critical renormalization fixed point for commuting pairs of area-preserving maps. The source code of our programs is available in the online version of this article at doi:10.1007/s00220-009-0922-1
The Institute of Electrical and Electronics Engineers, Inc., IEEE Standard for Binary Floating–Point Arithmetic. ANSI/IEEE Std 754–1985, New York: IEEE, 1985
Taft, S.T., Duff, R.A.: (eds), Ada 95 Reference Manual: Language and Standard Libraries, International Standard ISO/IEC 8652:1995(E), Lecture Notes in Computer Science 1246, New York: Spriger Verlag, 1999. See also http://www.adahome.com/rm95/
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Communicated by G. Gallavotti
This work was supported in part by MIUR project “Equazioni alle derivate parziali e disuguaglianze funzionali”.
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Arioli, G., Koch, H. The Critical Renormalization Fixed Point for Commuting Pairs of Area-Preserving Maps. Commun. Math. Phys. 295, 415–429 (2010). https://doi.org/10.1007/s00220-009-0922-1
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DOI: https://doi.org/10.1007/s00220-009-0922-1